My sister asked me something like five years ago to prove that a particular physical problem has a closed form solution. Are there some theorems to prove the existence of closed form solutions? The problem seems to be difficult as often you can make different kind of equations as differential- or integral- or recursive equations for some problems, but here we need a method to prove that no matter we define the equation, we can or can't find the closed form.

The equation appears in http://biology.anu.edu.au/hosted_sites/kokko/Esdale/index.html and https://physics.stackexchange.com/questions/63674/is-there-a-closed-form-solution-to-the-esdale-river-problem .

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    $\begingroup$ This question seems much too broad. There are existence theorems for certain kinds of differential equations, as well as integral equations. The problem to which you linked might result in a differential equation for which you cannot guarantee a solution. Existence can be a hard thing to prove for many physical problems. $\endgroup$ Aug 14, 2013 at 19:25

2 Answers 2


I really doubt there is for a general problem. Galois Theory gives a theorem which determines whether a polynomial has a solution in a closed form:

Theorem : $f(x)$, a polynomial, is soluble by radical $\Longleftrightarrow$ Gal$(f)$ is a soluble group.

Soluble by radical means that it can be expressed using the *,/,+,- and $n^\text{th}$ root which I think is what you mean by closed form. Knowing if Gal$(f)$ is soluble is a very hard work for easy functions and about impossible for the rest!

I know this doesn't exactly answer your question but knowing this, you should understand the reason why I say that I seriously don't think that there is such an algorithm or theorem.

  • $\begingroup$ Well, I was looking something bigger theorem which gives Galois theory and Risch's algorithm as a special case. $\endgroup$ Aug 14, 2013 at 19:34

The non-mathematical problems have to be translated to the corresponding mathematical problems.

A closed-form expression is a mathematical expression that contains only finite numbers of symbols and operations from a given set.
A mathematical problem is a closed-form problem if its solution is sought as a closed-form expression.

Trivially, every problem can be made to a closed-form problem - if the solution of that problem will be allowed.

Until now, there is no unified complete theory of closed-form solutions.

First, there are different kinds of mathematical problems: What general kinds of closed-form problems are there?
Maybe all these problems could be unified to one general mathematical problem someday: Solving systems of differintegral equations could unify the mathematical problems listed there.

Second, if a problem can be solved in closed form depends on the given set of allowed symbols and operations. And the applicable methods depend on this set.

But there are methods for single classes of functions and problems.

Solutions of algebraic equations by radicals are dealt by Galois theory and solutions of differential equations are dealt by Differential Galois theory. Maybe Differential Galois theory contains Galois theory.
Liouvillian solutions, among them elementary solutions, are dealt by Differential algebra. Risch structure theorem for algebraic dependence of Liouvillian functions [Risch 1979] is used for deciding the invertibility of Liouvillian functions by Liouvillian functions and for deciding the integrability of Liouvillian functions by Liouvillian functions (Risch algorithm), among them the elementary functions.

Risch structure theorem is applied by specifying a tower of differential fields for representig a class of functions. See e.g. section 1 of [Davenport 2007]. There are structure theorems like Risch's for other classes of functions:
Maybe they can be used for deciding invertibility and integrability in further classes of functions someday.
Maybe the method of towers of fields could be developed to a unified complete theory of classes of functions someday. Today, limitations exist due to the limitation in determining the algebraic dependence of transcendental numbers in the general case.
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[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Davenport 2007] Davenport, J. H.: What Might "Understand a Function" Mean? In: Kauers, M.; Kerber, M., Miner, R.; Windsteiger, W.: Towards Mechanized Mathematical Assistants. Springer, Berlin/Heidelberg, 2007, 55-65


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